3318 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 8, AUGUST 2010Fig. 1. Uniﬁlar diagram of the 3-phase system. Fig. 5. Cross sections used to represent the variation of relative position. C. Methodology: 2-D FE Model Transposition In this way, the cable performance is evaluated using 2-D ﬁnite element models coupled to the transposed multiple seg- ments technique (transposition) . In the particular case of UC, the interchanges are obtained performing various 2-D steady- state analyses considering several cross sections to model the different relative positions between the power cables and the metallic components. D. Least Common Multiple (LCM)Fig. 2. Cross-sections number 1 of the conﬁguration A and B. As described in Section II, the inner layer of this UC com- pletes 1 turn every 1000 mm while its outer layer completes 1 turn every 1500 mm. The LCM between the layer lengths is then 3000 mm. In other words, there is a repetition of the cable pat- tern every 3000 mm. E. Accuracy of the Analysis Method The number of cross sections deﬁnes the accuracy which is linked to the LCM of the cable layers, as shown in Fig. 5. For this UC one has compared the results for two values of cross sections number ( and ) and the difference was not signiﬁcant. Fig. 7(a) presents 3 of the 50 cross sections used to compute the simulations of the conﬁguration A and B.Fig. 3. Cross-sections number 1 of the conﬁguration C. F. 2-D Finite Element Analysis From the electromagnetism theory, the induced voltage in one conductor increases proportionally with the level and the fre- quency of the current of the surrounded conductors. The con- verters supply the oil pumps with a constant relation between voltage and frequency. Considering the operation characteris- tics, one of the worst conditions is when 3 circuits #1, #2 and #3 supply the pumps on 80 Hz and 1 circuit #4 supplies on 30 Hz. The general idea of the methodology is summarized in Fig. 6.Fig. 4. Transmission Lines phase transposition. The computational package FLUX-2-D  is used to evaluate each cross section the conﬁgurations.the variation of the relative position between the metallic cable IV. UMBILICAL CABLE ANALYSIScomponents. In order to compare the performance of the UC with the eval-B. Transposition Technique uation of the coupling effects between the internal components, three different conﬁgurations are veriﬁed: The concept of transposition is addressed in classical 3-phase • Conﬁguration A: the tubes and the power circuits turnhigh voltage transmission lines (TL) theory , . The idea is around the cable center, but the power circuits do not turnto change the position of phases through the TL length, aiming around their own centers, as in Fig. 7(a).to obtain balanced values for the TL parameters (inductances, • Conﬁguration B: the tubes and the power circuits turncapacitances and so on). The TL parameters are then calculated around the cable center and the power circuits turn alsofrom the average values on each phase. Fig. 4 illustrates a TL around their own centers, completing 1 turn every 750transposition with three interchanges with lengths L/3. mm, as shown in Fig. 7(b).
SALLES et al.: ELECTROMAGNETIC ANALYSIS OF SUBMARINE UMBILICAL CABLES 3319 Fig. 8. Magnetic potential lines obtained for the cross section 1 of the Conﬁg- uration A and B (only the ﬁrst cross section has the same values). (a) 80 Hz, (b) 30 Hz.Fig. 6. General steps of the methodology. Fig. 9. Real components of induced voltage (80 Hz) in phase C of circuit #4—Conﬁguration A.Fig. 7. Cross sections along the length of conﬁgurations A (a) and B (b). • Conﬁguration C: the tubes and the power circuits turn around the cable center (Fig. 3). Due to the magnetic permeability of the armor ( u0),the boundary conditions were deﬁned considering a tangentialmagnetic ﬁeld in the outer diameter of the UC. The supply cir-cuits are modeled with ideal 3-phase voltage sources and the oilpumps are modeled as a pure resistive load of eight ohms oneach phase. The circuit coupling represents each ﬁnite elementregion (power conductors, power shields, tubes and armor) byan electric circuit component.A. Terminal Voltage Computation Fig. 10. Imaginary components of induced voltage (80 Hz) in phase C of circuit Fig. 8(a) shows the equipotential lines obtained from the anal- #4—Conﬁguration A.ysis of the circuits #1, #2 and #3 supplied on 80 Hz. Fig. 8(b)shows the analysis of circuit #4 supplied on 30 Hz. These twoseparately analyses are necessary to determine the inﬂuence of Fig. 12. The induced voltage in conﬁguration C (not showed)the 80 Hz on the 30 Hz circuits for frequency domain simulation. has similar performance of conﬁguration A.The converter voltages of each phase A of the 3-phase circuitsare at its maximum value. B. Power Quality Analysis The average values of the real and the imaginary componentsof terminal voltage of the circuit #4 are calculated considering Voltage modulation indicates the induced voltage level in aeach cross section. The terminal voltage of the conﬁguration A phase of a circuit by the surrounding circuits. The computationand B obtained by the combined methodology are shown from of the induced voltage on phase A of circuit as a result ofFigs. 9 to 12. current ﬂowing through circuit can be calculated individually The induced voltages are almost constant for the conﬁgura- by the (1), (2) and (3)tion A, as shown in Fig. 9 and in Fig. 10. For the conﬁgurationB, the rotation of the 3-phase power circuits around their owncenters results on a variable induced voltage along the length (1)of the UC which compensates itself, as shown in Fig. 11 and in
3320 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 8, AUGUST 2010 TABLE II SIMULATION DATA OF DIMENSIONS AND MATERIAL PROPERTIESFig. 11. Real components of induced voltage (80 Hz) in phase C of circuit Conﬁguration B is the most appropriate one which guarantee#4—Conﬁguration B. the balanced voltage at the load terminals and the low level of modulation on the most sensible circuit (30 Hz). V. CONCLUSION The combined methodology presented in this paper enables the evaluation of the power quality of UC using 2-D steady- state models, without compromising the accuracy of the results. One has veriﬁed that the rotation of the power circuits results in a reduction of the voltage modulation. These characteristics will also guarantee a better oil pump operation, diminishing the maintenance and enlarging the lifetime. This analysis gives faster results compared to 3-D steady- state or 2-D transient. Besides that, the results of the 2-D tran- sient analysis (not showed in this paper) for the conﬁguration B are very similar to the one obtained by the frequency domain analysis. Even thought this methodology mainly analyses the in- duced voltage in the circuit #4 by the others 3 circuits, different analysis could also be done modifying the last two steps of theFig. 12. Imaginary components of induced voltage (80 Hz) in phase C of circuit Fig. 6, for example, to “loss computation” or “thermal analysis”.#4—Conﬁguration B. APPENDIX TABLE I The main data of the cable geometry and the material prop- VOLTAGE MODULATION IN THE 30 HZ CIRCUIT (%) erties are given in Table II. ACKNOWLEDGMENT The authors gratefully acknowledge the ﬁnancial support from the Brazilian government via FAPESP (State of São Paulo Research Foundation), CNPq (National Counsel of Technological and Scientiﬁc Development) and CAPES (“Co- ordenação de Aperfeiçoamento de Pessoal de Nível Superior,” in Portuguese). Computing total rms voltage on phase A of circuit , deﬁning as load terminal voltage at fundamentalfrequency of circuit REFERENCES  P. Pettersson and N. Schönborg, “Reduction of power system magnetic ﬁeld by conﬁguration twist,” IEEE Trans. Power Del., vol. 12, no. 4, (2) pp. 1678–1683, Aug. 1997.  C. Demoulias, D. P. Labridis, P. S. Dokopoulos, and K. Gouramanis, Voltage modulation in phase A of circuit , is “Ampacity of low-voltage power cables under nonsinusoidal currents,” IEEE Trans. Power Del., vol. 22, no. 1, pp. 584–594, Feb. 2007.  F. de Leon and G. J. Anders, “Effects of backﬁlling on cable ampacity (3) analyzed with the ﬁnite element method,” IEEE Trans. Power Del., vol. 23, no. 2, pp. 537–543, Apr. 2008.  W. D. Stevenson, Elements of Power System Analysis. New York: The induced voltage on the oil pump terminal of circuit #4 McGraw-Hill, 1982.  C. R. Paul, Analysis of Multiconductor Transmission Lines. Newby the 80 Hz circuits were computed and compared with its York: Wiley-Interscience, 1994.terminal voltage feed on 30 Hz. Table I presents the modulation  FLUX-2-D—Finite Element Package for Electromagnetic Computa-values in the circuit #4. tions, [Online]. Available: http://www.cedrat.com